Cayley-Menger Determinant
This entry contributed by Karen D. Colins
A determinant that gives the volume of a simplex in j dimensions. If S is a j-simplex in
with vertices
and
denotes the
matrix given by
(1)
then the content
is given by
(2)
where
is the
matrix obtained from
by bordering
with a top row
and a left column
. Here, the vector
L2-norms are the edge lengths and the
determinant in (2) is the Cayley-Menger determinant (Sommerville 1958, Gritzmann and Klee 1994). The first few coefficients for j = 0, 1, ... are -1, 2, -16, 288, -9216, 460800, ... (Sloane's
A055546).
For j = 2, (2) becomes
(3)
which gives the area for a plane triangle with side lengths a, b, and c, and is a form of Heron's formula.
For j = 3, the content of the 3-simplex (i.e., volume of the general tetrahedron) is given by the determinant
(4)
where the edge between vertices i and j has length
. Setting the left side equal to 0 (corresponding to a
tetrahedron of volume 0) gives a relationship between the
distances between vertices of a planar
quadrilateral (Uspensky 1948, p. 256).
Buchholz (1992) gives a slightly different (and slightly less symmetrical) form of this equation.
免责声明：本文为网络用户发布，其观点仅代表作者个人观点，与本站无关，本站仅提供信息存储服务。文中陈述内容未经本站证实，其真实性、完整性、及时性本站不作任何保证或承诺，请读者仅作参考，并请自行核实相关内容。
